Development of an 80 Gbit/s InP-based Mach-Zehnder Modulator

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Development of an 80 Gbit/s InP-based

Mach-Zehnder Modulator

vorgelegt von

M. Sc. Haitao Chen

aus Nanjing (V.R. China)

Von der Fakultät IV - Elektrotechnik und Informatik

der Technischen Universität Berlin

zur Erlangung des akademischen Grades

Doktor der Ingenieurwissenschaften

- Dr.-Ing. -

genehmigte Dissertation

Promotionsausschuss:

Vorsitzender: Prof. Dr. rer. nat. G. Tränkle

Berichter: Prof. Dr.-Ing. K. Petermann

Berichter: Prof. Dr. rer. nat. D. Jäger (Uni Duisburg-Essen)

Tag der wissenschaftlichen Aussprache: 26.10.2007

Berlin 2007

D 83

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Dedication

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Acknowledgements

First of all I would like to thank Professor Klaus Petermann and Dr. Herbert Venghaus for their supervision, inspiration, and guidance throughout all of this work. It has certainly been satisfying and rewarding to work with the modulator research group in Heinrich-Hertz-Institut. I offer my thanks to the members of the group for their long-term valuable help. Special thanks to Dr. Karl-Otto Velthaus and Dr. Detlef Hoffmann who helped me in this work. I would like to thank them also for their valuable advices, support, encouragement and valuable comments in writing of this dissertation and the published papers. Also, thanks to Dr. Ronald Kaiser, Holger Klein, Giorgis G. Mekonnen, Garbriele Bader, Bärbel Reinsperger, and Martin Gravert for their valuable help.

I would like also to acknowledge DAAD (the German Academic Exchange Service) and CSC (the China Scholarship Council), who give me the chance and financial support to accomplish my doctor study in Germany.

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Abstract

In this work, integrated InP-based electro-optical Mach-Zehnder modulators with capacitively loaded traveling-wave electrode are developed and characterized. The π phase shift in the upper and lower arm of the Mach-Zehnder interferometer is generated by electro-refraction due to the Quantum Confined Stark-Effect in a 20 period InGaAsP/InP multi quantum well (MQW) layer stack, which ensures a low driving voltage.

This thesis focuses on the microwave design and the optimization of the modulator. The basic theory that used in design and optimization are presented at first. Three different models are introduced and used for optimization. The significant design parameters of a capacitively loaded traveling wave electrode Mach-Zehnder modulator are analyzed in detail. Several optimizing strategies are given for reducing the microwave loss and improving the velocity match.

The experimental results suggest a packaged modulator has a 3 dBeo

bandwidth of 45.6 GHz and the driving voltage of 2.6 V with -2.8 V bias. The bandwidth achieves 57 GHz after optimizing the p-contact and decreasing the n-bulk resistance in the modulator, as well as improving the velocity matching by inserting a BCB buffer layer under the traveling wave electrodes. Measurements show that the packaged modulator module has a clear open NRZ eye diagram at 80 Gbit/s data rate.

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List of Tables

Table 1.1: Comparison of several types of optical intensity modulators...15

Table 4.1 Important physical properties of III-V compound materials [3]...74

Table 4.2 Calculated material properties for modulator modeling in HFSS ...75

Table 4.3 Material properties of the p-region with different built-in p-contact resistivity...76

Table 4.4 Calculated resistances in the active area of the modulator ...89

Table 4.5 Comparison of parameters before and after optimizing ...112

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List of Figures

Fig. 1.1 A transmitter in a 40 Gbit/s ETDM system...2

Fig. 1.2 Transition function of a current modulated semiconductor laser [2] ...4

Fig. 1.3 Direct modulation scheme ...4

Fig. 1.4 External modulation scheme ...5

Fig. 1.5 Schematic of a directional coupler ...6

Fig. 1.6 Schematic of a Mach-Zehnder interferometer with 2 Y-branches ...7

Fig. 1.7 Measured electro-optical response of a modulator...17

Fig. 2.1 CPW geometry and corresponding electric and magnetic field distribution [1] ...21

Fig. 2.2 CPS geometry and corresponding electric and magnetic field distribution [1] ...22

Fig. 2.3 Asymmetric CPS ...22

Fig. 2.4 Calculated characteristic impedance as function of different geometry parameters of CPS using semi-isolating InP as substrate ...24

Fig. 2.5 Voltage and current definitions and equivalent circuit for an incremental length of transmission line ...26

Fig. 2.6 A two-port network model for the transmission line...28

Fig. 2.7 A periodically loaded transmission line ...29

Fig. 2.8 Equivalent circuit of a unit cell in Fig. 2.7, uniform transmission line with shunt element in center ...29

Fig. 2.9 Lattice constant and bandgap energies of ternary and quaternary compounds formed by using III-V group semiconductors [10] ...32

Fig. 2.10 p-n junction under (a) reverse and (b) forward bias ...34

Fig. 2.11 Distributed resistance model of a contact to a thin semiconductor layer ...39

Fig. 2.12 (a) A TLM test structure and (b) contact resistance versus contact spacing ...40

Fig. 2.13 Illustration of index ellipsoid, the dashed ellipse is the intersection of the index ellipsoid with the plane normal to the propagation direction...41

Fig. 2.14 Energy levels of a QW structure without and with applied electric field [16] ...44

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Fig. 3.1 Schematic of a Mach-Zehnder Interferometer with 2 Y-branch ...49

Fig. 3.2 Transfer curve for MZI with 2 Y-branches ...50

Fig. 3.3 Transfer curve of an MZI with 2 Y-branches with optical loss, b=0.9 ...51

Fig. 3.4 Schematic of a MZM with a lumped electrode ...52

Fig. 3.5 Cross section of the optical waveguide covered with a lumped electrode ...52

Fig. 3.6 The equivalent circuit model for the MZM with a lumped electrode ...52

Fig. 3.7 Schematic of a MZM with push-pull lumped electrodes ...54

Fig. 3.8 The circuit model for series push-pull [4] ...54

Fig. 3.9 Schematic of a MZI with one Y-branch and a 2×2 MMI and the ideal transfer matrix for each section...54

Fig. 3.10 Transmission character of Mach-Zehnder interferometer ...56

Fig. 3.11 Intensity modulation using push-pull structure ...56

Fig. 3.12 Schematic of a MZM with capacitive loaded segmented traveling-wave electrode...58

Fig. 3.13 Cross sectional view of a MZM with DC bias and HF circuit diagram ...58

Fig. 3.14 Microwave design of the capacitively loaded TWE MZM ...60

Fig. 3.15 Equivalent circuit of a traveling-wave electrode modulator ...61

Fig. 4.1 Flow chart of general design process of HFSS...72

Fig. 4.2 Flow chart of the HFSS adaptive refinement process ...73

Fig. 4.3 Simplified p-layer in modulator modeling in HFSS...76

Fig. 4.4 Carrier concentration of different layers in modulator with varied bias [7]...77

Fig. 4.5 Top-view of a whole chip modulator model with 12 sections in HFSS...78

Fig. 4.6 (a) A section of modulator (b) An N-section segmented modulator ...78

Fig. 4.7 An 8-cell 2-port network in Ansoft Designer...79

Fig. 4.8 (a) Top-view (b) Side-view of a single-section modulator model in HFSS...80

Fig. 4.9 A 3-D view of a single-section modulator model with excitation and boundary condition ...80

Fig. 4.10 Comparison between measured and simulated electrical insertion loss S21 of the same modulator structure ...81

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Fig. 4.13 Transmission matrix representation of TWE MZM ...84

Fig. 4.14 Calculated L0 and C0 for an unloaded CPS...85

Fig. 4.15 Measured bias dependent capacitance of circular p-i-n diodes ...86

Fig. 4.16 SEM photo of inner electrodes of the modulator ...87

Fig. 4.17 Parasitic capacitance in the active area of a modulator ...88

Fig. 4.18 The resistances in the active area of a modulator...89

Fig. 4.19 The hierarchical model for one section of a modulator in an EDA tool ...90

Fig. 4.20 Circuit models of passive part and active part in a section of the modulator ...90

Fig. 4.21 12-section TWE MZM model in ADS ...91

Fig. 4.22 The simulation results using the detailed model ...92

Fig. 4.23 Measured S21 parameters of two modulators (l=3 mm, η=0.4) with 250 μm and 500 μm period length, respectively...93

Fig. 4.24 The propagation of optical and electrical signals in the modulator [14] ...95

Fig. 4.25 Measured data of a 12.2 mm long waveguide by the Fabry-Perot method ..97

Fig. 4.26 Measured S11 and converted corresponding nμ of a modulator (#45C63) ....98

Fig. 4.27 Measured phase and converted corresponding nμ of a modulator (#45C63) 99 Fig. 4.28 Microwave index and the optic index of two modulators ...100

Fig. 4.29 Cross-sectional view of CPS test structures ...101

Fig. 4.30 Microwave indices of unloaded TWE test structures...101

Fig. 4.31 Electro-optical response of the modulators with and without BCB layer ..102

Fig. 4.32 The electro-optical responses of a modulation under co- and counter- propagation conditions, respectively ...103

Fig. 4.33 The CPS structures simulated in HFSS ...105

Fig. 4.34 Simulated insertion loss of 1 mm CPS ...105

Fig. 4.35 CPS with mesa...106

Fig. 4.36 The impact of mesa width on the insertion loss of the CPS ...107

Fig. 4.37 The impact of the series resistance in the active area on the insertion loss of the modulator ( for Rc=10.7 Ω, the corresponding ρc is 3×10-5 cm2·Ω )...108

Fig. 4.38 Simulated insertion loss of modulators with different ρc using HFSS ...110

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Fig. 4.40 The relationship between the 3 dBeo bandwidth, the driving voltage and the

fill factor of a modulator ...114

Fig. 4.41 Measured C-V curves with two different i-region thickness...115

Fig. 4.42 The measured Vπ before and after optimizing i-layer’s thickness...116

Fig. 4.43 Loading capacitance as function of microwave index of the unloaded CPS ...116

Fig. 4.44 Cross sectional view of the modulator (a) before and (b) after redesign ...117

Fig. 4.45 Characteristic impedance for CPS as a function of aspect ratio a/b...118

Fig. 4.46 Flow chart of the redesign process ...119

Fig. 5.1 SEM photo of a pair of air bridges between TWE and optical waveguide in a fabricated modulator ...124

Fig. 5.2 SEM photo of one side of a fabricated modulator...124

Fig. 5.3 (a) A 3-D model of the spot size converter with field distribution for the fundamental mode in the passive and fiber port waveguide, respectively (b) Alignment tolerances measured at the input port of the modulator compared to that of the tolerances of a tapered fiber to a passive waveguide [1] ...125

Fig. 5.4 The CAD drawing of a packaged modulator module...126

Fig. 5.5 Measured electrical loss S21 of a modulator before and after packaging ...127

Fig. 5.6 A simplified 3-D model in HFSS ...128

Fig. 5.7 Top-view and side-view of the 3-D model of a packaged module and simulation results ...129

Fig. 5.8 Bonding wires (a) two bonding wires for each connection, (b) five bonding wires for ground and three for signal...130

Fig. 5.9 An optimized integrated CPS-CPW transition in a modulator...131

Fig. 5.10 Three different configurations of input and output pads (a) asymmetrical S-G configuration, (b) symmetric S-S-G configuration, (c) S-G-S-S-G pads, (d) measured electrical insertion losses of the three different configurations ...131

Fig. 5.11 The module using G-S-G pads with long ground bar on the middle block 132 Fig. 5.12 The module using G-S-G pads with separated middle blocks...133

Fig. 5.13 Simulated electrical insertion loss S21 of a modulator before and after optimized packaging ...133

Fig. 6.1 The circular diode test structure: top view and cross sectional view. ...134

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Fig. 6.3 Intrinsic layer thickness over bias voltage deduced from C-V measurements

...135

Fig. 6.4 Layout of one TLM structure (wc=100 μm) ...136

Fig. 6.5 Measured p-contact resistivities of two modulator wafers...137

Fig. 6.6 Setup of DC matrix measurement [2]...139

Fig. 6.7 (a) Normalized optical transmission with applied voltage to I- and U-electrode. (b) The transfer curve of the corresponding operating point in (a)...140

Fig. 6.8 The setup of electrooptic response measurement (200 MHz – 65 GHz) [3] 143 Fig. 6.9 Setup for 40 Gbit/s eye-diagram measurements of modulator ...145

Fig. 6.10 Setup for 80 Gbit/s eye-diagram measurements of modulator ...146

Fig. 7.1 A packaged modulator module [1] ...149

Fig. 7.2 Normalized output vs. driving voltage of the module (#45 A50) at -2.8 V bias ...149

Fig. 7.3 EO response of the packaged MZI modulator (#45 A50) at -2.8 V bias [1] 150 Fig. 7.4 Measured NRZ eye diagram of the packaged MZI modulator (#45 A50) at 40 Gbit/s with 1.55 µm optical signal (PRBS word length: 231-1, 10 ps/div) [1]...150

Fig. 7.5 Measured 40 Gbit/s PRBS BER curves of the packaged modulator (#45A50) ...151

Fig. 7.6 Measured NRZ eye diagram of the packaged MZI modulator (#45 A50) at 80 Gbit/s with 1.55 µm optical signal (PRBS word length: 215-1, 5ps/div) ...152

Fig. 7.7 Electro-optical response of a modulator (on-chip) with a TWE length of 4 mm. (lE= 200 µm, Λ= 500 µm, -6 V bias)...153

Fig. A.1 An arbitrary N-port network [1] ...155

Fig. A.2 A two-port network...156

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List of Symbols

α Microwave attenuation constant

αc attenuation constant due to Ohmic loss

αchirp chirp parameter

αd attenuation constant due to dielectric loss

βμ complex microwave propagation constant of the signal in the electrode

βopt complex optical propagation constant of the signal in the optical waveguide

μ

β_opt velocity mismatch in the modulator

γ complex transmission constant

δ skin effect factor

εreff effective dielectric constant

η fill factor of a modulator

λ wavelength

λ0 wavelength in free space

Λ period length of a modulator

μe electron mobility

μh hole mobility

ρ1 input electrode structure reflection coefficient

ρ2 output electrode structure reflection coefficient

ρc contact resistivity

σ conductivity

ωm modulation frequency

φ(f) accumulated phase delay of S21

a lattice constant

b optical imbalance factor in MZI C0 shunt capacitance per unit length

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CL loading capacitance of a modulator per unit length Cpin junction capacitance of the p-i-n diode per unit area Cs parasitic capacitance per unit length

d distance between two optical waveguides db diameter of the air-bridge

di thickness of intrinsic layer dp thickness of the p-layer

dpin depletion layer width of reverse biased pin diode ds skin depth

Eg bandgap energy fBragg Bragg frequency

G0 shunt conductance per unit length l traveling-wave electrode length L0 series inductance per unit length Lb inductance of the air-bridge lb length of the air-bridge lE active electrode length leff effective modulation length n electron density

n0 optical index of the active layer at zero applied voltage nopt optical index

nμ microwave index

nμ’ microwave index of the loaded CPS p hole density

Pin input optical power Pout output optical power

r modulation reduction factor R0 series resistance per unit length Rc contact resistance

rij electro-optic coefficient Rn bulk resistance of the n-layer

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Rp bulk resistance of the p-layer Rsh sheet resistance

tanδ dielectric loss tangent

TMZI optical intensity transfer function vµ velocity of the electrical signal Vg amplitude of the generator voltage vgr-o group velocity of the optical signal vopt velocity of the optical signal vp phase velocity

Vπ driving voltage wc width of the contact

wwg width of the optical waveguide

Z0 characteristic impedance of unloaded CPS Zm characteristic impedance of the modulator ZS source impedance, normally is 50 Ω ZT terminating impedance

Δλ wavelength difference between two adjacent maximum points

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List of Abbreviations

3-D Three dimensional

ASE Amplified spontaneous emission BCB Benzocyclobuten

BER Bit error rate

CPS Coplanar strips CPW Coplanar waveguides C-V Capacitance-Voltage

CW Continuous wave

DWDM Dense wavelength division multiplexing EA Electroabsorption

EAM Electroabsorption modulator ECL External cavity laser

EDFA Erbium-doped fiber amplifier EM Electromagnetic EO Electro-optic

ETDM Electrical time division multiplexing FEM Finite element method

GPIB General purpose interface bus

G-S Ground signal

G-S-G Ground signal ground

III-V Semiconductor based on group III and group V elements MBE Molecular beam epitaxy

MMI Multimode interference

MOCVD Metal organic chemical vapor deposition MQW Multi quantum wells

MZI Mach-Zehnder interferometer MZM Mach-Zehnder modulator NLO Nonlinear optical

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PRBS Pseudorandom binary sequence QCSE Quantum confined Stark effect SEM Scanning electron microscope

S-G Signal ground

SOA Semiconductor optical amplifier SOLT Short, open, load, thru

SSC Spot-size converter TDM Time division multiplexing TE Transverse electric TEM Transverse electromagnetic TLM Transmission line method

TM Transverse magnetic TWE Traveling-wave electrode

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Chapter 1 Introduction

1.1 OVERVIEW

The use of light for communication purposes is not new. It can date back to antiquity if we interpret optical communications in a broad sense [1]. Most civilizations have used mirrors, fire beacons, or smoke signals to convey a single piece of information [2].

Development of fibers and devices for optical communication began in the early 1960s and continues strongly today. But the real change came in the 1980s. During this decade optical communication in public communication networks developed from the status of a curiosity into being the dominant technology. Transmitting signals by using light through optical fiber is the most effective way to move large amounts of data rapidly over long distances. Nowadays, in the era of information technology, the optical fiber transmission links form the backbone of the communications infrastructure. Recently, the bit rates of the electrical time-domain multiplexed (ETDM) system are improving from 2.5 Gbit/s to 10 Gbit/s and even 40 Gbit/s in deployed systems.

It is well known that the electrical signal still conquers our communication terminals in everyday life. But to take advantage of the optical transmission system, these electrical signals must be converted into optical ones which can be transferred in fibers. The electro-optic modulator (cf. Fig. 1.1) is such a device that converts the electrical signal to the optical one. For example, in a 40 Gbit/s electrical time-division multiplexing (ETDM) transmitter, low speed signals used in communication terminals will be hierarchically multiplexed into a high speed one (40 Gbit/s) at first, and then

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the high speed electrical signal will be transformed into an optical signal which can be transferred in an optic fiber as shown in Fig. 1.1.

Fig. 1.1 A transmitter in a 40 Gbit/s ETDM system

In addition, high-bandwidth optical modulation has also applications in fiber-optic RF signal transmission and optoelectronic signal processing [3].

Lightwaves have various characteristics that can be modulated to carry information, including the intensity, phase, frequency and polarization. Among these, the intensity modulation is the most popular for optical fiber communication systems, primarily due to the simplicity of envelope photodetection. The modulator discussed in this work also uses intensity modulation.

1.2 ORGANIZATION OF THE DISSERTATION

This dissertation is organized as follows. In Chapter 1, different types of modulators are introduced and compared. The motivation of this work is presented at the end of the chapter.

In Chapter 2 the theoretical foundation of traveling-wave electrode electro-optic modulators is presented. Chapter 3 discusses the design consideration of the high-speed Mach-Zehnder modulator. Chapter 4 describes the electrical optimization to 80 Gbit/s including the simulations and experiments.

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Chapter 5 explains the fabrication process and the involved packaging technology.

Chapter 6 introduces the corresponding measurements that are used for characterizing the modulators and test structures.

Chapter 7 shows experimental results and gives an overall summary and an outlook of further work.

1.3 DIRECT MODULATION

There are two groups of modulation methods: one is direct modulation, the other is external modulation. Each has its own pros and cons. They will be introduced in this and next section in detail.

The most straightforward method for modulation is to directly modulate the laser source. Due to the requirements of bandwidth and efficiency, only semiconductor lasers are of practical interest for direct modulation [4].

A unique feature of semiconductor lasers is that the semiconductor laser can be modulated directly by modulating the excitation current. As shown in Fig. 1.2 for a semiconductor laser, the output power of light intensity increases linearly with the injection current above threshold. For an injection current I = I0 + i(t), the optical output power is P(t) = P0 + p(t), where (I0,P0) is the bias point for the direct modulation of the semiconductor laser [5]. Using this feature, the direct modulation scheme can be configured as Fig. 1.3.

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Fig. 1.2 Transition function of a current modulated semiconductor laser [2]

Fig. 1.3 Direct modulation scheme

Modulation bandwidths of 25 GHz and higher have been reported for semiconductor lasers operating at 1.55 μm [5] and shorter wavelengths. The widest 3 dB bandwidth of a directly modulated laser reported to date is 37 GHz [6]. However, when the modulation frequency increases toward the relaxation resonance frequency of a semiconductor laser, both the relative intensity noise and distortions increase rapidly [7]. This severely limits the feasibility of direct modulation for a higher frequency. The large frequency chirp also precludes the direct modulation for long-distance communications systems (this will be explained in Chapter 2).

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1.4 EXTERNAL MODULATORS

At bit rates of 10 Gbit/s or higher, the frequency chirp imposed by direct modulation becomes large enough that direct modulation of semiconductor laser is rarely used [2].

Unlike direct modulation, external modulation has been shown to have superior performance for wide bandwidth optical fiber communications, however with the potential disadvantages of adding system complexity and cost.

Fig. 1.4 External modulation scheme

As shown in Fig. 1.4 for a high-speed transmitter, the laser is biased at a constant current to provide the CW output, and an external optical modulator placed next to the laser converts the CW light into a data-coded pulse train with correct modulation format.

In the external modulation scheme, the light output passes through a material whose optical properties can be modified by an applied external electric field.

External optical modulators with bandwidths greater than 10 GHz are typically fabricated from the electro-optic crystal LiNbO3, electro-optic polymers or III-V

semiconductor compounds. Some of these devices have shown modulation frequencies in excess of 40 GHz.

According to the physical effects that are used in the respective modulator, these modulators can be grouped into two categories: one is the electro-optic (EO) modulator, the other is the electro-absorption (EA) modulator.

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1.4.1 Electro-Optic Modulators

The electro-optic effect is most widely used for high speed modulation. Electro-optic modulators can produce amplitude, frequency or phase modulation in an optical signal by exploiting the electro-optic effect in which the optical properties of a crystal can be altered by an electric field.

Directional coupler and Mach-Zehnder interferometer are two waveguide structures that are often used for intensity modulation with the electro-optic effect.

1.4.1.1 Directional coupler

A directional coupler consists of two side by side waveguides separated by a few micrometers (cf. Fig. 1.5). The overlap of guided waves in the two waveguides couples energy back and forth between the waveguides.

Optical Input Guide1 Guide2 Bar State Cross State

Fig. 1.5 Schematic of a directional coupler

When no voltage is applied, the two guides are phase-matched. The light input into guide 1 is nearly 100% coupled into guide 2 (cross-state) provided the coupler length is properly chosen.

When a voltage is applied, there is a change in the refractive index due to the applied electric field which can be made unequal in the two guides by implementing an appropriate design. This leads to a mismatch in the phase velocities in both guides and also to a reduction in coupling length. By adjusting the device parameters most of

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the power can be transferred back to guide 1 (the bar-state) at the end of the directional coupler.

For directional coupler devices based on bulk material, the size of the device is fairly large since the electro-optic coefficient is not so large. Typically, the devices require an interaction length of ~1cm or more and a bias voltage of ~5-10V. The switching voltage of a directional coupler modulator is 3 times larger than that required for a Mach-Zehnder modulator (see section 1.4.1.2). Due to these reasons, for digital applications the directional coupler structure is less been used than the Mach-Zehnder interferometer in intensity modulator design.

However, recent research shows that with multi-section design directional coupler based modulators have high linearity which is important for analog applications, such as RF-fiber links. Several corresponding designs have been published [12,13].

1.4.1.2 Mach-Zehnder Interferometer

A Mach-Zehnder optical modulator uses an interferometric configuration of waveguides to enable the conversion of phase modulation into intensity modulation. Fig. 1.6 is a schematic drawing of such an interferometer. The optical wave enters from the left side and then splits equally into two beams by a splitter. The divided two beams propagate in two separate waveguides that are isolated from each other. The waveguides are often called “two arms”. At the end two beams are combined together.

arm1 arm2

combiner

splitter

ΔΦ

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The splitter and combiner here can be a Y-branch, a directional coupler or a 2×2 multimode interference (MMI) coupler.

A relative phase difference Δφ is introduced between two arms. When the differential phase shift Δφ between the two arms equals ± , destructive interference π occurs, corresponding to the off-state or “0” level for the modulator. With no differential phase shift, constructive interference occurs, corresponding to the on-state or “1” level for the modulator.

Due to its simple structure and the potential of push-pull operation (further discussed in Chapter 3), the Mach-Zehnder interferometer (MZI) is the most popular device for implementing optical intensity modulation using the electro-optic effect. The modulators in this work are also based on the concept of the MZI.

1.4.1.3 Materials

Many different materials can be used to fabricate electro-optic modulators, including lithium niobate, III-V compounds, polymers and even silicon. In this section, a brief introduction will be given for each material mentioned above.

Lithium Niobate (LiNbO3)

Lithium niobate (LiNbO3) is the most widely used material for the

manufacture of electro-optic devices, including phase modulators, polarization modulators, Mach-Zehnder intensity modulators and directional-coupler intensity modulators.

LiNbO3 is a ferroelectric anistropic crystal with 3-m crystal symmetry.

Depending on the orientation of an applied electric field, different electrooptic coefficients are relevant.

The most popular method of fabricating optical waveguides in LiNbO3 is

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the waveguide [8]. The dimensions of the optical waveguide can be controlled by properly choosing the initial Ti stripe width, film thickness and diffusion conditions. The resulting single-mode waveguide has a very low optical propagation loss, typically less than 0.2 dB/cm, and its mode size can be matched very well to that of a single-mode optical fiber.

Due to its properties of enabling low-loss waveguides, high electro-optic effect and high optical coupling efficiency with single-mode optical fiber, so far LiNbO3 has

been the material of choice for optical modulators at bit rates of 10 Gbit/s and above. LiNbO3 travelling-wave modulators, based on a Mach-Zehnder interferometer

waveguide structure, are the most widespread modulators in deployed systems [8]. However, LiNbO3 Mach-Zehnder modulators have not only a large size, but

also a bias-drifting problem which requires an extra bias control circuit. High driving voltage also limits their applications. In addition, the LiNbO3 modulator is difficult to

be integrated with other components

III-V Semiconductors

III-V compound semiconductors, such as GaAs, InP and their ternary and quaternary alloys, are also candidate materials for EO modulators. There are two types of applications: one is using bulk material, the other is based on multi-quantum-wells (MQW).

Although their EO coefficients (defined in section 2.6) are 20 times smaller than that of LiNbO3, efficient modulation can still be obtained with these materials.

This is because semiconductor crystal growth and fabrication techniques provide great flexibility for waveguide geometry control, so that the optical guided mode can be confined to a very small region (2-3 μm spot size), and thus a very large electric field can be achieved even with a small voltage applied across the small dielectric gap. Additionally, III-V semiconductors have large optical refractive indices, for example,

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compared to 2.2 for LiNbO3. This indicates a 3-4 times improvement for the index

change in a linear electro-optic modulation application. All these factors make the efficiency of III-V EO modulators comparable to that of LiNbO3 modulators. In

addition, III-V EO modulators can potentially be integrated with a wide range of components such as lasers, semiconductor optical amplifiers (SOAs), photodetectors, passive optical circuits and even electronic drivers.

One popular type of III-V semiconductor EO modulator is based on bulk GaAs/AlGaAs waveguides grown on GaAs substrates due to the availability of large GaAs substrates (4" diameter), and those EO modulators are typically 2-3 cm long. In the GaAs/AlGaAs waveguide, the GaAs layer has higher optical index than the AlGaAs layer, the latter is lattice-matched to GaAs for all values. By sandwiching the GaAs layer between two AlGaAs layers, optical confinement in the vertical growth direction can be achieved. The lateral optical confinement is usually obtained by material etching. Shallow-etched rib waveguides are preferred over deep-ridge waveguides to achieve single-mode waveguiding. The resulting optical waveguides may have propagation loss ranging from tenths of dB/cm to a few dB/cm, depending on the waveguide structure and the fabrication process.

Another type of modulator, which is also made of III-V compound materials, uses the quantum confined Stark effect (QCSE) in multi-quantum-wells (MQW). Since strong index changes can be induced by QCSE (this will be discussed in detail in section 2.7), the relevant driving voltage for this type of device is relatively low compared to that of LiNbO3 modulators. This is very attractive, because the lower the

driving voltage is, the lower the power consumption, cost and size of the transmitter module become [9]. In addition, the compact size is also a notable advantage. The modulators in this work are also of this type.

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Polymers

Compared to LiNbO3 and III-V semiconductors, organic polymers are

relatively immature EO materials, but they also possess a great potential for more advanced modulators. For example, a polymer modulator operating up to 150 GHz was reported in 2002 [10].

The advantages of EO polymers mostly come from the applicability of the spin-coating technique. This does not only make it possible to integrate polymer EO devices with various electronic and optoelectronic components [11], but also offers the opportunity to fabricate multiple devices stacked in the vertical direction. Metal electrodes can be buried between different polymer layers, and this makes the electrode design very flexible. To lower the fabrication cost, polymer EO devices can be fabricated directly on top of optical submounts. The optical refractive index of polymers is close to that of single-mode optical fibers. This provides a good match between the polymer waveguide mode and the fiber mode.

Polymer modulators may have great potential in the future, but their optical power handling capability is currently much less than that of modulators based of other materials, and their thermal and photochemical stabilities need to be improved.

Silicon

Unlike the materials mentioned before, silicon as a ‘new’ material of optical components has been prompted on account of the demand for low-cost solutions from industry. There are also strong motivations for considering optics for shorter distance interconnects, especially using silicon photonic, a platform in which one could make electronics, waveguides, other optical components and optoelectronic devices.

Silicon is attractive from a cost standpoint because mature silicon processing technology and manufacturing infrastructure already exist and can be used to build

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provides the possibility of monolithically integrating optical elements and advanced electronics on silicon using bipolar or CMOS technology [12].

Although unstrained pure crystalline silicon exhibits no linear electro-optic effect and the refractive index changes due to the Franz-Keldysh and Kerr effects are also small [13], several silicon waveguide-based optical modulators have been proposed and demonstrated [14,15] by using the free carrier plasma dispersion effect in a forward-biased pin diode geometry.

Since the modulation speed due to the free carrier plasma dispersion effect is determined by the rate at which carriers can be injected and removed, the long recombination carrier lifetime in the intrinsic silicon region generally limits the modulation frequency of these silicon-based devices.

A breakthrough was claimed by a group from Intel Corp. in 2004. They made a Si-based electro-optical modulator that can modulate light at 1~2 GHz, which is unprecedented in a silicon-based modulator, by introducing a metal-oxide-semiconductor (MOS) capacitor as phase shifter into the modulator structure [16]. Because charge transport in the MOS capacitor is governed by majority carriers, the device bandwidth is not limited by the relatively slow carrier recombination processes of pin diode devices. This group had optimized the device for a 10 Gbit/s application, recently [17].

Another breakthrough in silicon modulators was announced in 2005 by a team from Stanford University. They have reported that QCSE in thin germanium quantum-well structures grown on silicon has strengths comparable to that in III-V materials [18]. This discovery may be promising for small, high-speed, modulator devices fully compatible with silicon electronics manufacture.

Although there have been claimed breakthroughs in silicon-based modulator design, currently the modulation bandwidth and the power consumption of such devices is still not comparable with that of modulators based of other materials.

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1.4.2 Electroabsorption Modulators

The electroabsorption (EA) effect denotes the change of the optical absorption coefficient in materials due to the presence of an electric field. The EA effect in a single optical waveguide directly results in optical intensity modulation. The primary materials for fabricating EAMs working at 1.3-1.6 μm optical wavelengths are currently III-V semiconductors, specifically, ternary and quaternary alloys (including InGaAs, InAlAs, InAsP, InGaP, InGaAsP and InGaAlAs, etc.) grown on InP substrate. Similar to semiconductor-based EO modulators, EAMs can also be integrated with a variety of other devices. The most notable integrated chips include electroabsorption modulated lasers (EML) [19], and tandem EA modulators (often with SOA) for RZ data transmission and wavelength-division demultiplexing [20]. Another attraction of EAMs is their compact size (typically 80-300 μm long) due to their high efficiency. This leads to small footprint for a single device and high yield per wafer which lowers the fabrication cost.

An EA waveguide usually consists of p-i-n semiconductor layers, among which the intrinsic layer has higher optical refractive index than the p-type and n-type doped layers. This provides vertical optical confinement. The lateral optical confinement is usually achieved by a deep mesa etch. The etched deep-ridge waveguide can then be planarized using polyimide or regrown semi-insulating InP. In a typical EA waveguide design, both the optical mode and the applied electric field are tightly confined in a small area around the intrinsic layer, which enables highly efficient modulation.

The optical propagation loss in EA waveguides is much larger than that in EO waveguides, typically ranging from 15 to 20 dB/mm. This high optical propagation loss is mainly attributed to three sources: the first is the residual absorption loss in the active layer; the second is the interband absorption loss induced by free carriers in the

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by the roughness of the deep-ridge waveguide sidewalls and defects in the grown materials. Fortunately, due to their short length the overall optical loss is tolerable.

The tight optical confinement results in a small, elliptical mode profile in EA waveguides, typically 2-4 μm lateral and 1-2 μm vertical effective mode sizes which can have large optical coupling loss to a single-mode optical fiber whose mode size is typically 9 μm in diameter. To overcome this problem, either a micro ball lens is inserted between the EA waveguide and the fiber tip, or an optical lens can be directly fabricated on the fiber tip to produce a mode size of 3-4 μm at the focal plane that is 20 μm away from the fiber tip. Integrating a spot-size converter (SSC) at both ends of the EA waveguide can further reduce the coupling loss. A SSC is a passive optical waveguide with adiabatic vertical and/or lateral tapers in geometry and/or optical refractive index. It converts the small elliptical EA waveguide mode at one end to a near-circular expanded mode (mode size 3-4 μm) at the other end, or vice versa. Integrating with a SSC makes the EAM longer, which makes cleaving and handling easier and also reduces the amount of unguided light directly coupled from the input fiber to the output fiber, resulting in an improved ON/OFF extinction ratio for digital links. Such SSC can be also used in a MQW based EO modulator.

Compared with EO modulators, the main disadvantages of EAM are low saturation power, large chirp, narrow optical bandwidth and need of temperature control. Nevertheless, EAMs are very attractive due to their small chip-size, low driving voltage and the compatibility for integration with other components. Limited by their inherent chirp, EAMs are not suitable for long-haul systems and are normally used in short distance transmission systems.

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1.4.3 Comparison of Different Types of External Modulators

In the previous sections different types of modulators based of different materials have been briefly introduced. Table 1.1 gives a short summary of those modulators with a comparison of a few important characteristics.

Table 1.1: Comparison of several types of optical intensity modulators

TYPE EO MZM EAM

III-V Compound MATERIAL LiNbO3

bulk MQW Polymer InP

Optical loss low medium medium low high Driving voltage high medium low high low

Chirp Low low low low high

Footprint big big small big small

Highest speed 50 GHz 40 GHz 57 GHz 110 GHz 50 GHz

Commercial use yes yes yes yes yes

extra Bias-control circuit

- - - Temperature

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1.5 MOTIVATION FOR THE WORK

The rapid growth of data traffic, caused mainly by increasing use of the Internet including recent expanded use of multimedia services and wireless access, has stimulated the demand for high-capacity photonic network systems. High capacity optical networks are based on a combination of wavelength-division multiplexing (WDM) and time-division multiplexing (TDM) for purposes ranging from long-distance transmission between nodes to short-reach optical links such as metro/access links. Moreover, the next generation networks (NGN) are not far away from us and the 100 Gbit/s Ethernet is also in progress. All of these will inevitably need transmitters which can operate above 40 Gbit/s.

The maximum direct modulation bandwidth in semiconductor lasers in limited to a few GHz. The frequency chirp associated with high-speed direct modulation of laser diodes above a few Gbit/s has become a clear problem and results in serious consequences in high data rate long-distance optical-fiber transmission systems. The electro-optical modulator is the only choice for transmitters operating at bit rates beyond 10 Gbit/s in a long-haul transmission system.

Nowadays, electro-optical modulators have become the decisive components in the ETDM long-haul transmission system, whose data speed exceeds 40 Gbit/s. There is no such a product with corresponding speed in the market. Although there are reports about 80 Gbit/s or 100 Gbit/s ETDM transmitter systems [21,22], the modulators used in those systems are still 40 Gbit/s ones, but not 80 Gbit/s or 100 Gbit/s ones. This largely restricts the performance of the whole system. The aim of this work is to develop an electro-optical modulator enabling operation well above 40 Gbit/s in order to improve the performance of the corresponding ETDM system.

Moreover, high speed modulation of optical waves is also of great interest for radar, satellite links and electronic warfare systems.

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As mentioned in section 1.4, the InP-based MQW Mach-Zehnder modulator has compact size, low driving voltage, zero or controllable chirp and the compatibility for integration with other components. In addition, it is also suitable for advanced modulation formats such as CS-RZ [23] and RZ-DPSK [24] for example. Its optical insertion loss could also be reduced to a level comparable to that in LiNbO3-based

modulators. All these properties make it attractive for modulation rates well above 40 Gbit/s.

The modulator research group in the Fraunhofer-Institute for Telecommunications Heinrich-Hertz-Institut has many years of research experience in InP-based MQW Mach-Zehnder modulators. The electro-optical response of a modulator of a preliminary version had achieved 20 GHz (cf. Fig. 1.7) and served as a basis for further development. This work focuses on optimizing the electrical design of the developed modulators and making them work up to 80 Gbit/s.

0 10 20 30 40 50 -9 -6 -3 0 e.o. S 21 [dB] Frequency [GHz] S21 (#36B9) 20GHz

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REFERENCES

[1] G. J. Holzmann and B. Pehrson, “The early history of data networks”, IEEE computer society press, Los Alsmitos, CA (1995)

[2] G. P. Agrawal, “Fiber-optic communication systems”, 3rd Ed., Wiley, New York (2002)

[3] R.G. Walker, “Low-voltage, 50 Ω, GaAs/AlGaAs travelling-wave modulator with bandwidth exceeding 25 GHz”, Electron. Lett. 25, 1549-1550 (1989). [4] S. L. Chuang, “Physics of Optoelectronic Devices”, Wiley, 487-488 (1995) [5] P. A. Morton, R. A. Logan, T. Tanbun-Ek, P. F. Sciortino, Jr, A. M. Sergent, R.

K. Montgomery, and B. T. Lee, “25 GHz bandwidth 1.55 μm GaInAsP p-doped strained multiquantum well lasers,” Electron. Lett. 28, 2156–2157 (1992). [6] S. Weiser, E. C. Larkins, K. Czotscher, w. Benz, J. Daleiden, J. Fleissner, M.

Maier, J. D. Ralston, B. Romero, A. Schonfelder and J. Rosenzweig, “ 37 GHz direct modulation bandwidth in short-cavity InGaAs/GaAs MQW lasers with C-doped active regions,” paper SCL1.1, IEEE Lasers and Electro-Optic Society Annual Meeting, San Jose, CA, Oct. 30- Nov. 2, (1995)

[7] N. K. Dutta, N. A. Olsson, L. A. Koszi, P. Besomi, R. B. Wilson, and R. J. Nelson, “Frequency chirp under current modulation in InGaAsP injection lasers,” J. Appl. Phys. 56, 2167-2169 (1984).

[8] K. Noguchi, O. Mitomi, and H. Miyazawa, “Millimeter-Wave Ti:LiNbO3

Optical Modulators, ” IEEE J. Lightwave Technology, 16, 615-619 (1998)

[9] T. Yamannka, “High-performance InP-based optical modulators”, OWC1, OFC2006

[10] M. Lee, H. E. Katz, C. E. D. M. Gill, P. Gopalan, J. D. Heber and D. J. McGee, “Broadband Modulation of light by using an electro-optic polymer,” Science,

298, 1401-1403 (2002)

[11] S. Kalluri, M. Ziari, A. Chen, V. Chuyanov, W. H. Steier, D. Chen, B. Jalali, H. Fetterman, and L. R. Dalton, “Monolithic integration of waveguide polymer electrooptic modulators on VLSI circuitry,” IEEE Photon. Technol. Lett. 8, 644-646 (1996)

[12] R. A. Soref, “Silicon-based optoelectronics,” Proc. IEEE 81, 1687-1706 (1993) [13] R. A. Soref and B. R. Bennett, “Electro optical effects in silicon,” IEEE J.

Quantum Electron, QE-23, 123-129 (1987)

[14] C. K. Tang and G. T. Reed, “Highly efficient optical phase modulator in SOI waveguides,” Electron. Lett. 31, 451-452 (1995)

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[15] P. Dainesi, “CMOS compatible fully integrated Mach-Zehnder interferometer in SOI technology,” IEEE Photon. Technol. Lett. 12, 660-662 (2000)

[16] A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal-oxide-semiconductor capacitor,” Nature 427, 615-618 (2004)

[17] L. Liao, “High-speed silicon Mach-Zehnder modulator,” Opt. Express 13, 3129 (2005)

[18] Y. Kuo, Y. Lee, Y. Ge, S. Ren, J. E. Roth, T. I. Kamins, D. A. B. Miller, and J. S. Harris, “Strong quantum-confined Stark effect in germanium quantum-well structures on silicon,” Nature 437, 1334-1336 (2005)

[19] K. Wakita, I. Kotaka, H. Asai, M. Okamoto, Y. Kondo, and M. Naganuma, “High-speed and low-drive voltage monolithic multiple quantum-well modulator/DFB laser light source,” IEEE Photon. Technol. Lett. 4, 16-18, (1992) [20] N. Souli, A. Ramdane, F. Devaux, A. Ougazzaden, and S. Slempkes, “Tandem

of electroabsorption modulators integrated with distributed feedback laser and optical amplifier for 20-Gbit/s pulse generation and coding at 1.5 μm,” in Proc. OFC’97 Tech. Dig. (Optical Fiber Communication), 6, paper WG6, 141-142 (1997)

[21] K. Schuh, “85.4Gbit/s ETDM transmission over 401 km SSMF applying UFEC”, ECOC2005 Th4.1.4, (2005)

[22] Peter J. Winzer, “107Gb/s optical ETDM transmitter for 100 G Ethernet transport”, ECOC2005 Th4.1.1, (2005)

[23] Y. Miyamoto, “320 Gbit/s (8x40 Gbit/s) WDM transmission over 367 km with 120 km repeater spacing using carrier-suppressed return-to-zero format,” Electronics Lett. 35, 2041-2042 (1999)

[24] A. H. Gnauck, “2.5 Tb/s (64x42.7 Gbit/s) transmission over 40x100 km NZDSF using RZ-DPSK format and all-Raman-amplified spans,” in Technical Digest of Optical Fiber Communication Conference (OFC2002) , Post-deadline paper, FC2 (2002).

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Chapter 2 Basic Theory

The traveling-wave electrode Mach-Zehnder modulator is an electro-optic device. To design and optimize such a device, it involves the knowledge of semiconductor, high frequency microwave, and electro-optical effects. All of these will be introduced briefly in this chapter.

2.1 COPLANAR LINES

The term coplanar lines is used for those transmission lines where all the conductors are in the same plane, namely on the top of the dielectric substrate. Coplanar waveguide (CPW) (cf. Fig. 2.1a) and coplanar strips (CPS) (cf. Fig. 2.2a) are coplanar lines [1]. A distinct advantage of these two types of transmission line lies in the fact that mounting lumped components in shunt or series configuration is much easier. Drilling holes or slots through the substrates is not needed.

(a) CPW geometry (b) Electric and magnetic field distribution Fig. 2.1 CPW geometry and corresponding electric and magnetic field distribution [1]

As shown in Fig. 2.1(a), CPW consists of two slots each of width w printed on a dielectric substrate. The spacing between the slots is denoted by s. The electric and magnetic field configurations for quasi-static approximation are shown in Fig. 2.1(b).

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The configuration of CPS, which is complementary to the CPW, is shown in Fig. 2.2(a). CPS consists of two strips, generally of equal width w on a dielectric substrate. The spacing between the strips is denoted by s. The electric and magnetic field configurations are shown in Fig. 2.2(b).

(a) CPS geometry (b) Electric and magnetic field distribution Fig. 2.2 CPS geometry and corresponding electric and magnetic field distribution [1]

Sometimes, it is also desirable to use an asymmetric CPS (cf. Fig. 2.3) to adjust the characteristic impedance by changing the width of one of the strips while keeping the width of the other strip and the slot width fixed.

Fig. 2.3 Asymmetric CPS

CPW and CPS support TEM modes and have been analysed using quasi-static as well as full-wave methods. Expressions for characteristic impedance Z0 and effective dielectric constant εreff of CPW, CPS and asymmetric CPS are given in table 2.1.

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Table 2.1 Expressions for coplanar lines characteristic impedance and effective dielectric constant [1]

Structure Characteristic Impedance (Ω) Effective dielectric constant

CPW 0 30 ( ) ( ) reff K k Z K k π ε ′ = CPS 0 120 ( ) ( ) reff K k Z K k π ε = ′ 1 1 1 '( ) ( ) 1 2 ( ) '( ) r reff K k K k K k K k ε ε = + − , , 2 2 a s s k a b w b = = = + 1 sinh( ) 2 sinh( ) 2 a h k b h π π = Asymmetric CPS 2 0 2 ( ) 60 ( ) reff K k Z K k π ε = ′ 1 2 2 1 2 1 w w k s w s w = − + +

w1 and w2 are the strips’ width

The K(k) in table 2.1 represents a complete elliptic function of the first kind and is defined as 2 2 2 0 ( ) 1 sin d K k k π _ϕ ϕ = − ⋅

∫

(2.1.1) ( )

K k′ is its complementary function and can be calculated by

2 2

( ) ( ) 1

K k +K k′ = (2.1.2)

Expression (2.1.3) [2] is an accurate and simple approximation of ( ) ( ) K k K k′ and has only a relative error lower than 3×10-6:

2 2 ( ) 2 for 0 ( ) 1 2 ln(2 ) 1 ( ) 1 1 2 ln(2 ) for 1 ( ) 1 2 with 1 K k k K k k k K k k k K k k k k π π ≈ ≤ ≤ ′ _⋅ + ′ ′ − + ≈ ⋅ ⋅ ≤ ≤ ′ − ′ + = (2.1.3)

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As an example, Fig. 2.4 shows a calculated characteristic impedance curve of the CPS using semi-isolating InP as the substrate.

0.05 0.10 0.15 0.20 0.25 0.30 0.35 30 40 50 60 70 80 90 100 Characte ri stic i m pe da nce [ Ω ] a/b

calculated Zo: CPS on S.I. InP ε_r=12.5

Fig. 2.4 Calculated characteristic impedance as function of different geometry parameters of CPS using semi-isolating InP as substrate (a, b defined in table 2.1)

The coplanar lines have three types of losses: dielectric, Ohmic and radiation/surface wave. The expression for the attenuation constant due to dielectric loss in CPS is [1] 0 1 tan 27.3 dB/unit length 1 reff r d r reff ε ε δ α ε λ ε − = ⋅ ⋅ ⋅ − (2.1.4)

Here λ0 is the wavelength in free space, tanδ is the dielectric loss tangent of the substrate material.

The Ohmic loss (also called conductor loss) of coplanar lines is given as [1]

1 1 2 1 1 1 1 1 8.68 ₁ ₈ ₍₁ ₎ ₁ ₈ ₍₁ ₎ ln( ) ln( ) 480 ( ) ( )(1 ) (1 ) (1 ) s reff c R _a _k _b _k K k K k k a t k b t k ε _π _π α π π π ⎧ ⎡ − ⎤ ⎡ − ⎤⎫ ⎪ ⎪ = _⎨ _⎢ + _⎥+ _⎢ + _⎥_⎬ ′ − _⎪_⎩ _⎣ + _⎦ _⎣ + _⎦_⎪_⎭ (2.1.5)

with unit of dB/unit length. Here a, b, εreff , k and k1 are defined in table 2.1, series expansions for K(k), needed for the evaluation of (2.1.5) are given as [3]

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(

)

2 3 4 2 2 2 2 2 4 6 for 0 0.707 ( ) 1 2 9 50 306.25 ... 2 8 8 8 8 for 0.707 1 9 7 25 37 ( ) 1 ... 2 4 6 2 4 30 2 4 where ln k k k k k K k k k k k K k p p p p p k π ≤ ≤ ⎧ _⎛ _⎞ _⎛ _⎞ _⎛ _⎞ _⎛ _⎞ ⎫ ⎪ ⎪ = _⎨ + _⎜ _⎟+ _⎜ _⎟ + _⎜ _⎟ + _⎜ _⎟ + _⎬ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎪ ⎪ ⎩ ⎭ ≤ ≤ ′ ′ ′ ⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞⎛ ⎞ = + − _{⎜ ⎟} + _⎜ − _{⎟⎜ ⎟} + _⎜ − _{⎟⎜ ⎟} + ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎛ ⎞ = ⎜ ′_⎝ ⎟_⎠ (2.1.6)

In addition to the dielectric and Ohmic losses, coupling of power to surface waves and radiation from unwanted (parasitic) modes also contribute to the total loss of coplanar lines. Coplanar lines with finite thickness substrate can also be leaky. However, if the substrate thickness h according to [4]

0 0.12 r h λ ε ⋅ < (2.1.7)

power leakage can be avoided since the cutoff frequency of the surface modes is pushed above the operating frequency. That is, for InP substrate and work frequency of 40 GHz, the substrate thickness h should be smaller than 255 μm.

2.2 TRANSMISSION LINE THEORY

At frequencies where the wavelength is several orders of magnitude larger than the greatest dimensions of the circuit or system being examined, conventional circuit elements such as capacitors, inductors, resistors and transistors are the basic building blocks for the circuits or systems. The description or analysis of such circuits may be adequately carried out in terms of loop currents and node voltages without consideration of propagation effects. However, as the frequency is raised to a point where the wavelength is no longer large compared to the circuit dimensions, propagation effects can no longer be ignored [5]. This problem can be solved by using

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transmission line theory. On the other hand, coplanar lines are also transmission lines. Hence, the transmission line theory is the basis for analyzing the high-speed traveling-wave modulator.

2.2.1 The Lumped-element Circuit Model of Transmission Line

A unique property of the transmission line is that a satisfactory analysis of its properties may be carried out by treating it as a network with distributed parameters and solving for the voltage and current waves that may propagate along the line. As shown in Fig. 2.5(a), a transmission line is often schematically represented as a two-wire line since transmission lines always have at least two conductors. The short piece of line of length Δz of Fig. 2.5(a) can be modeled as a lumped-element circuit, as shown in Fig. 2.5(b), where R0, L0, G0, C0 are per unit length quantities of series resistance, series inductance, shunt conductance and shunt capacitance, respectively.

Δz + -v(z,t) i(z,t) L0Δz G0Δz R0Δz C0Δz + -v(z+Δz,t) i(z+Δz,t)

(a) voltage and current definitions (b) lumped-element equivalent circuit Fig. 2.5 Voltage and current definitions and equivalent circuit for an incremental

length of transmission line

The series inductance L0 represents the total self-inductance of the two conductors, and the shunt capacitance C0 is due to the close proximity of the two conductors. The series resistance R0 represents the resistance due to the finite conductivity of the conductors, and the shunt conductance G0 is due to the dielectric loss in the material between the conductors. R0 and G0, therefore, represent loss. A finite length of transmission line can be viewed as a cascade of sections of the form of Fig. 2.5(b).

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According to Kirchhoff’s law and letting Δz Æ 0, we can get the following relationships 0 0 ( ) ( ) ( ) dV z R j L I z dz = − + ω (2.2.1) 0 0 ( ) ( ) ( ) dI z G j C V z dz = − + ω (2.2.2) with ( , ) ( ) j t v z t =V z eω _(2.2.3) ( , ) ( ) j t i z t =I z eω _(2.2.4)

2.2.2 Propagation Parameters of the Transmission Line

The propagation equations of a transmission line can be obtained from (2.2.1) and (2.2.2): 2 2 2 d V V dz =γ _(2.2.5) 2 2 2 d I I dz =γ _(2.2.6)

with the complex transmission constant γ given by

0 0 0 0

(R j L G)( j C ) j

γ = + ω + ω = +α β (2.2.7)

Traveling-wave solutions for (2.2.5) and (2.2.6) are

0 0 z z V V e−γ V eγ + − = + _(2.2.8) 0 0 z z I =I e₊ −γ +I e₋ γ _(2.2.9)

where the _e−γz_{term represents wave propagation in the +}_{z direction, and the}_eγz

term represents wave propagation in the -z direction, V₀_± and I₀_± are constants that should be defined by the boundary conditions.

The phase velocity is given by

p

v ω

β

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The characteristic impedance Z0 of the transmission line can be defined as [5] 0 0 0 0 0 R j L Z G j C ω ω + = + (2.2.11)

As shown in Fig. 2.6, the transmission line with characteristic impedance Z0, propagation constant γ, and length l can be described as a two-port network. The transmission matrix (also called ABCD matrix, definition in Appendix A) of this two-port network is [6] 0 0 cosh( ) sinh( ) 1 sinh( ) cosh( ) l Z l A B l l C D Z γ γ γ γ ⎡ ⎤ ⎡ _{⎤ ⎢} _⎥ = ⎢ _{⎥ ⎢} _⎥ ⎣ _{⎦ ⎢} _⎥ ⎣ ⎦ (2.2.12)

Fig. 2.6 A two-port network model for the transmission line

One can get the transmission matrix by converting the measured scattering matrix as follows [6] 11 22 12 21 11 22 12 21 21 21 11 22 12 21 11 22 12 21 21 21 (1 )(1 ) (1 )(1 ) 2 2 (1 )(1 ) (1 )(1 ) 1 2 2 r r S S S S S S S S Z S S A B C D S S S S S S S S Z S S + − + + + − ⎡ ⎤ ⎢ ⎥ ⎡ _{⎤ ⎢} _⎥ = ⎢ ⎥ _⎢ ₋ ₋ ₋ ₋ ₊ ₊ _⎥ ⎣ ⎦ ⎢ ⎥ ⎣ ⎦ (2.2.13)

here Zr is the reference impedance used in the measurement and is normally equal to 50 Ω.

The propagation parameter γ and the characteristic impedance Z0 can be calculated from (2.2.12): ln( ) 2 A D BC l γ + ₊ = (2.2.14)

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0 B Z

C

= (2.2.15)

From (2.2.7) and (2.2.11), the R0, L0, G0, C0 can be calculated by (2.2.16) using the propagation parameter γ and characteristic impedance Z0, respectively.

0 0 0 0 0 0 0 0 Im( ) R = Re( ) Im( ) G = Re( ) γ _γ ω γ γ ω ⋅ = ⋅ = Z L Z Z C Z (2.2.16)

2.2.3 Analysis of the Periodically Loaded Transmission Line

The periodically loaded transmission line (cf. Fig 2.7) may have a lower phase velocity than the smooth transmission line. This property is used as a slow-wave property in modulator design.

Fig. 2.7 A periodically loaded transmission line

Fig. 2.8 Equivalent circuit of a unit cell in Fig. 2.7, uniform transmission line with shunt element in center

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The transmission matrix for a unit cell as Fig. 2.8 can be described as:

0 0

cosh sinh cosh sinh

1 0

2 2 2 2

1

1 1

sinh cosh sinh cosh

2 2 2 2 l l l l Z Z A B C D l l Y l l Z Z γ γ γ γ γ γ γ γ ⎡ ⎛ ⎞ ⎛ ⎞⎤ ⎡ ⎛ ⎞ ⎛ ⎞⎤ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎢ ⎥ ⎢ ⎥ ⎡ ⎤₌_⎢ ⎝ ⎠ ⎝ ⎠_⎥_⋅⎡ ⎤_⋅_⎢ ⎝ ⎠ ⎝ ⎠_⎥ ⎢ ⎥ _⎢ _⎛ _⎞ _⎛ _⎞ _⎥ ⎢ ⎥ _⎢ _⎛ _⎞ _⎛ _⎞ _⎥ ⎣ ⎦ ⎣ ⎦ ⎢ ⎜_⎝ ⎟_⎠ ⎜_⎝ ⎟_⎠ ⎥ ⎢ _⎝⎜ ⎟_⎠ _⎝⎜ ⎟_⎠ ⎥ ⎣ ⎦ ⎣ ⎦ (2.2.17)

Multiplication of the matrices and use of identities for hyperbolic functions gives [7]

( )

(

)

( )

(

)

( )

0 0 0 0 0 0 0 0 0 cosh sinh 2 1 cosh( ) sinh 2 1 1 cosh( ) sinh 2 YZ A D l l YZ B Z l l YZ C l l Z γ γ γ γ γ γ = = + ⋅ ⎡ ⎤ = ⋅_⎢ ⋅ − + + _⎥ ⎣ ⎦ ⎡ ⎤ = ⋅_⎢ ⋅ + + _⎥ ⎣ ⎦ (2.2.18)

Note that the unit cell is symmetric (A₀ =D₀) and reciprocal (A D₀ ₀−B C₀ ₀ = ). 1 The overall transfer matrix for N like networks is just the Nth power of that for a single network: 0 0 0 0 N A B A B C D C D ⎡ ⎤ ⎡ ⎤ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ (2.2.19)

For a network with reciprocity, this results in [7]

0

0 0 0

0 0

sinh sinh( 1) sinh

sinh sinh sinh sinh( 1) sinh sinh sinh with cosh 2 A N N N B A B C D N D N N C A D ⋅ Γ − − Γ Γ ⎡ _⋅ ⎤ ⎢ ⎥ ⎡ ⎤_{= ⎢} Γ Γ _⎥ ⎢ ⎥ _Γ _⋅ _{Γ −} _{− Γ} ⎣ ⎦ ⎢ _⋅ ⎥ ⎢ _Γ _Γ ⎥ ⎣ ⎦ + Γ = (2.2.20)

Because of the unit cell’s symmetry and reciprocity, the matrix of whole network can be simplified as [7] 0 0 0 0 0 cosh sinh sinh cosh with cosh B N N C A B C D _C N N B A ⎡ ⎤ Γ Γ ⎢ ⎥ ⎡ _{⎤ ⎢} _⎥ = ⎢ _{⎥ ⎢} _⎥ ⎣ _{⎦ ⎢} _Γ _Γ _⎥ ⎢ ⎥ ⎣ ⎦ Γ = (2.2.21)

Development of an 80 Gbit/s InP-based Mach-Zehnder Modulator